[1] 194.0844 197.4556
Chinstrap Penguins
Nov 23, 2024
Image Source: Gemini AI
Y_i|\mu \sim N(\mu, \sigma^2)
Image Source: wikipedia.org
\mu \sim N(193.7, {6.12}^2)
\mu|(Y = 68) \sim N(?,?) \ \sigma = 7.13, \ n = 68, \ \tau = 6.12, \ \bar{y} = 193.37
\mu | \overset{\to}y \sim N\left( 195.82 \frac{7.13 ^2}{68*6.12^2 + 7.13^2} + 193.37\frac{68*6.12^2}{68*6.12^2 + 7.13^2}, \frac{6.12^2 7^2}{68*6.12^2 + 7.13^2} \right)\\
\text{Simplified: N} \left( 195.82 * 0.01957 + 193.37 * 0.9804, 0.7329 \right)\\
\text{N: } (193.42, 0.856^2)
| Model | Mean | Mode | Variance | SD |
|---|---|---|---|---|
| Prior | 193.37 | 193.37 | 37.4544 | 6.12 |
| Posterior | 195.7721 | 195.7721 | 0.7329711 | 0.8561373 |
95\% \text{ CI} = [194.08, 197.46]
Image Source: commons.wikimedia.org
Chinstraps: 215 and 225 mm
Null Hypothesis H_0:
Alternative Hypothesis H_1:
Image Source: Gemini AI
Image Source: crittersquad.com
[1] 0
[1] 0
P(215\le\mu\le225|Y=68)=0
This confirms the conclusion drawn previously.
Image Source: redbubble.com
[1] 0.0002042996
[1] 0.0002043414
[1] 0
\text{Bayes Factor}= \frac{\text{Posterior Odds}}{\text{Prior Odds}}= \frac{0}{0.0002}=0
| Hypotheses | Prior Probability | Posterior Probability |
|---|---|---|
| H_0: \mu \in (215,225) | P[H_0] = 0.9998 | P[H_0 \mid Y = 68] = 1 |
| H_1: \mu \notin (215,225) | P[H_1] = 0.0002 | P[H_1 \mid Y = 68] = 0 |
Image Source: Gemini AI